Unified Foundations for Mathematics

TL;DR

This paper proposes the theory of named sets as a unified foundational framework for mathematics, addressing the limitations of set theory and other foundational approaches.

Contribution

It introduces the theory of named sets as a comprehensive and fundamental foundation that unifies various mathematical theories and generalizations.

Findings

Named sets serve as a universal foundation for mathematics.

The theory encompasses and generalizes traditional set theory and its variants.

It offers a potential solution to the foundational diversity in mathematics.

Abstract

There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole mathematics. Set theory has been for a long time the most popular foundation. However, it was not been able to win completely over its rivals: logic, the theory of algorithms, and theory of categories. Moreover, practical applications of mathematics and its inner problems caused creation of different generalization of sets: multisets, fuzzy sets, rough sets etc. Thus, we encounter a problem: Is it possible to find the most fundamental structure in mathematics? The situation is similar to the quest of physics for the most fundamental "brick" of nature and for a grand unified theory of nature. It is demonstrated that in contrast to physics, which is still in search for a unified theory, in mathematics such a theory exists. It is the theory of named sets.